We now transfer this result to the context of Lie groups.
Let \(A\)  be a Banach algebra and \(\pi \!:G\rightarrow {\mathop {\mathrm {GL}}\nolimits }(A)\)  a holomorphic homomorphisms. (Here \({\mathop {\mathrm {GL}}\nolimits }(A)\)  is the group of invertible elements of \(A\) , which can be treated as a Banach-Lie group; see, e.g., [1]}).
We can identify the Lie algebra of \({\mathop {\mathrm {GL}}\nolimits }(A)\)  with \(A\)  [2]}. Applying the Lie functor to \(\pi \)  we get a Banach-Lie algebra homomorphism \(\mathrm {L}_\pi \!: \mathfrak {g} \rightarrow A \)  such that \(\pi \exp =\exp \mathrm {L}_\pi \) , where \(\exp \)  denotes the exponential maps \(\mathfrak {g}\rightarrow A\)  and \(A\rightarrow {\mathop {\mathrm {GL}}\nolimits }(A)\) .
